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Abstract

Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let $Γ: X → 2^{Φ}$ be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), $Γ(x)=∂^{-α}_{Φ}f|_{x}$.